Group #46782916 — subgroups

Subgroups — #46782916

Order 33664.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22554403	C₂	1	2	yes	maximal
2	C₂×#22554401	C₂	1	2	yes	maximal
3	C₂×#22554381	C₂	1	1	yes	maximal, characteristic
4	#30681808	C₂	1	1	yes	maximal, characteristic
5	#382×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₆₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₃	—	2	2	no	—
8	C₂×C₄⋊Dic₂₆₃	C₂²	1	2	yes	—
9	#22554403	—	2	4	no	—
10	#22554401	—	2	4	no	—
11	#22554381	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
15	#95×C₂₆₃	—	2	2	no	—
16	C₂²×Dic₂₆₃	—	2	2	no	—
17	C₂×C₂₆₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₂₆₃⋊C₈	D₄	1	4	yes	—
19	C₂×C₂₆₃⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₂₆₃	—	2	2	no	—
21	C₄⋊Dic₂₆₃	—	2	2	no	—
22	C₄⋊Dic₂₆₃	—	4	4	no	—
23	C₂²×C₄×C₂₆₃	—	2	1	no	—
24	C₂²×C₄×C₂₆₃	—	2	1	no	—
25	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₃	—	2	2	no	—
28	C₄⋊C₄×C₂₆₃	—	2	2	no	—
29	C₄²×C₂₆₃	C₂×C₄	1	2	yes	—
30	C₄²×C₂₆₃	Q₈	1	2	yes	—
31	C₂×Dic₂₆₃	—	4	2	no	—
32	C₂×Dic₂₆₃	—	4	4	no	—
33	C₂₆₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
34	C₂³×C₂₆₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₆₃	—	2	2	no	—
36	C₂×C₄×C₂₆₃	—	2	2	no	—
37	C₂×C₄×C₂₆₃	—	4	1	no	—
38	C₂×C₄×C₂₆₃	—	4	1	no	—
39	C₂×C₄×C₂₆₃	—	4	2	no	—
40	C₂×C₄×C₂₆₃	—	4	2	no	—
41	C₂×C₄×C₂₆₃	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₆₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₆₃	#127	1	1	yes	characteristic
48	C₂²×C₂₆₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₂₆₃	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₂₆₃	#137	1	2	yes	—
51	C₄×C₂₆₃	—	2	4	no	—
52	C₄×C₂₆₃	—	8	1	no	—
53	C₄×C₂₆₃	—	8	1	no	—
54	C₄×C₂₆₃	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₂₆₃	C₄⋊Q₈	1	2	yes	—
56	Dic₂₆₃	—	8	2	no	—
57	C₂×C₂₆₃	#384	1	1	yes	characteristic
58	C₂×C₂₆₃	#428	1	2	yes	—
59	C₂×C₂₆₃	#496	1	1	yes	characteristic
60	C₂×C₂₆₃	#497	1	1	yes	characteristic
61	C₂×C₂₆₃	#501	1	2	yes	—
62	C₂₆₃	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	263	1	no	maximal, Sylow
64	C₂×#107	—	263	2	no	—
65	C₂×#106	—	263	2	no	—
66	C₂×#97	—	263	1	no	—
67	#384	—	263	1	no	—
68	#382	D₂₆₃	1	1	yes	characteristic
69	C₂²×C₈	—	263	2	no	—
70	C₂×C₄⋊C₄	—	263	2	no	—
71	C₂×C₄⋊C₄	—	526	2	no	—
72	C₂×C₄⋊C₄	C₂×D₂₆₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₂₆₃	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
75	#107	—	526	4	no	—
76	#106	—	526	4	no	—
77	#97	—	263	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	526	2	no	—
82	C₂²×C₄	C₂²⋊D₂₆₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
84	C₂×C₈	—	263	4	no	—
85	C₂×C₈	—	263	4	no	—
86	C₂×C₈	—	263	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	526	2	no	—
90	C₄⋊C₄	—	526	2	no	—
91	C₄⋊C₄	—	1052	4	no	—
92	C₄²	C₄×D₂₆₃	1	2	yes	—
93	C₄²	Dic₅₂₆	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	1052	2	no	—
102	C₂×C₄	—	1052	4	no	—
103	C₂×C₄	Dic₂₆₃⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₂₆₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₂₆₃	1	2	yes	—
106	C₂×C₄	Q₈×D₂₆₃	1	2	yes	—
107	C₂×C₄	C₂×Dic₅₂₆	1	1	yes	characteristic
108	C₈	—	263	8	no	—
109	C₂²	C₂×Dic₂₆₃⋊C₄	1	1	yes	characteristic
110	C₂²	#22554466	1	1	yes	characteristic
111	C₂²	#22554468	1	1	yes	characteristic
112	C₂²	#22554495	1	2	yes	—
113	C₂²	#22554500	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	2104	2	no	—
118	C₄	C₄⋊C₄×D₂₆₃	1	2	yes	—
119	C₄	#22554469	1	2	yes	—
120	C₂	#30681807	1	1	yes	characteristic
121	C₂	#30681981	1	2	yes	—
122	C₂	#30682328	1	1	yes	characteristic
123	C₂	#30682333	1	1	yes	characteristic
124	C₂	#30682348	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22554403

C₂

yes

maximal

C₂×#22554401

C₂

yes

maximal

C₂×#22554381

C₂

yes

maximal, characteristic

#30681808

C₂

yes

maximal, characteristic

#382×C₂₆₃

C₂

yes

maximal, characteristic

C₂²×C₂₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₆₃

—

C₂×C₄⋊Dic₂₆₃

C₂²

yes

—

#22554403

—

#22554401

—

#22554381

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₆₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₃

C₂²

yes

characteristic

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

#95×C₂₆₃

—

C₂²×Dic₂₆₃

—

C₂×C₂₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₆₃⋊C₈

D₄

yes

—

C₂×C₂₆₃⋊C₈

Q₈

yes

—

C₄⋊Dic₂₆₃

—

C₄⋊Dic₂₆₃

—

C₄⋊Dic₂₆₃

—

C₂²×C₄×C₂₆₃

—

C₂²×C₄×C₂₆₃

—

C₂²×C₄×C₂₆₃

D₄

yes

—

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₃

—

C₄⋊C₄×C₂₆₃

—

C₄²×C₂₆₃

C₂×C₄

yes

—

C₄²×C₂₆₃

Q₈

yes

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂₆₃⋊C₈

C₄⋊C₄

yes

complemented

C₂³×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂×C₄×C₂₆₃

C₂×Q₈

yes

—

C₂×C₄×C₂₆₃

C₂×Q₈

yes

characteristic

C₂²×C₂₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₃

#127

yes

characteristic

C₂²×C₂₆₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₆₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₆₃

#137

yes

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₆₃

C₄⋊Q₈

yes

—

Dic₂₆₃

—

C₂×C₂₆₃

#384

yes

characteristic

C₂×C₂₆₃

#428

yes

—

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

—

C₂₆₃

#1847

yes

characteristic, complemented, Sylow

#1847

—

263

maximal, Sylow

C₂×#107

—

263

—

C₂×#106

—

263

—

C₂×#97

—

263

—

#384

—

263

—

#382

D₂₆₃

yes

characteristic

C₂²×C₈

—

263

—

C₂×C₄⋊C₄

—

263

—

C₂×C₄⋊C₄

—

526

—

C₂×C₄⋊C₄

C₂×D₂₆₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₆₃

yes

characteristic

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

#107

—

526

—

#106

—

526

—

#97

—

263

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

526

—

C₂²×C₄

C₂²⋊D₂₆₃

yes

—

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

C₂×C₈

—

263

—

C₂×C₈

—

263

—

C₂×C₈

—

263

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

526

—

C₄⋊C₄

—

526

—

C₄⋊C₄

—

1052

—

C₄²

C₄×D₂₆₃

yes

—

C₄²

Dic₅₂₆

yes

—

C₂³

C₂×C₂²⋊D₂₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

1052

—

102

C₂×C₄

—

1052

—

103

C₂×C₄

Dic₂₆₃⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₂₆₃

yes

characteristic

105

C₂×C₄

D₄×D₂₆₃

yes

—

106

C₂×C₄

Q₈×D₂₆₃

yes

—

107

C₂×C₄

C₂×Dic₅₂₆

yes

characteristic

108

C₈

—

263

—

109

C₂²

C₂×Dic₂₆₃⋊C₄

yes

characteristic

110

C₂²

#22554466

yes

characteristic

111

C₂²

#22554468

yes

characteristic

112

C₂²

#22554495

yes

—

113

C₂²

#22554500

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

2104

—

118

C₄

C₄⋊C₄×D₂₆₃

yes

—

119

C₄

#22554469

yes

—

120

C₂

#30681807

yes

characteristic

121

C₂

#30681981

yes

—

122

C₂

#30682328

yes

characteristic

123

C₂

#30682333

yes

characteristic

124

C₂

#30682348

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₆₃
commutator	C₂×C₄×C₂₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	#382×C₂₆₃	C₂
socle, prime radical	C₂³×C₂₆₃	C₂×D₄
cosocle	C₂²×C₄×C₂₆₃	C₂³
hypercenter	#382	D₂₆₃
hypocenter	C₂₆₃	#1847
LCS	G ▷ C₂×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₆₃

commutator

C₂×C₄×C₂₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₃

Fitting

#382×C₂₆₃

C₂

socle, prime radical

C₂³×C₂₆₃

C₂×D₄

cosocle

C₂²×C₄×C₂₆₃

C₂³

D₂₆₃

C₂₆₃

G ▷ C₂×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
11. #22554381	C₄	semidirect
33. C₂₆₃⋊C₈	C₄⋊C₄	semidirect
62. C₂₆₃	#1847	semidirect

H ≅ G/N

type

11. #22554381

C₄

semidirect

33. C₂₆₃⋊C₈

C₄⋊C₄

semidirect

62. C₂₆₃

#1847

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22554403	2
C₂×#22554401	2
C₂×#22554381	2
#30681808	2
#382×C₂₆₃	2
#1847	263

subgroup

index

#382×C₂₆₃

263

Subgroups — #46782916

Conjugacy classes of subgroups

Complements

Maximal subgroups